Paciﬁc Journal of Mathematics - MSP · PACIFIC JOURNAL OF MATHEMATICS Vol. 175, No. 1, 1996 SMALL EIGENVALUE VARIATION AND REAL RANK ZERO OLA BRATTELI AND GEORGE A. ELLIOTT A necessary

PacificJournal ofMathematics

SMALL EIGENVALUE VARIATION AND REAL RANK ZERO

OLA BRATTELI AND GEORGE A. ELLIOTT

Volume 175 No. 1 September 1996

PACIFIC JOURNAL OF MATHEMATICS

Vol. 175, No. 1, 1996

SMALL EIGENVALUE VARIATION AND REAL RANK ZERO

OLA BRATTELI AND GEORGE A. ELLIOTT

A necessary and sufficient condition, in terms of asymp-totic properties of the sequence, is given for the inductivelimit of a sequence of finite direct sums of matrix algebrasover commutative C*-algebras to have real rank zero (i.e., foreach self-adjoint element to be approximible by one with finitespectrum).

1. Introduction.

In this paper we will consider C*-algebra inductive limits A = limAn where

the C*-algebras An have the form

(1.1) A* = 0 C(Ωnti, M,.) ,i=i

with each Ωn> Am(Λχ) obtained from Φ m ? n by cutting down by theunit of Am(Aι) in Am and restricting to An(A). Let

/P(Ωn) denote the setof partitions of Ωn into clopen sets which are refinements of the partition

{Ωn,i, Ωn>2, ,ΩnΓτι} .

47

48 OLA BRATTELI AND GEORGE A. ELLIOTT

For any compact metrizable space Ω, let dimΩ denote the covering di-mension of Ω; see [Eng]. We will assume throughout that dimΩn < H-oofor each n. (Our results also hold without this restriction, but in a vacuousway.)

In [BBEK] and [BDR] it was proved that A has real rank zero, i.e., thatany self-adjoint element in A can be approximated by self-adjoint elementswith finite spectrum, in several situations. For instance, this was proved in[BBEK] under the conditions that A be simple and unital, each Ω,nj be theunion of a finite number of connected clopen subsets, dim Ωn < 2 for all n,and the projections of A separate the tracial states of A. By using techniquesfrom [DNNP], the condition that dimΩn < 2 in this result was replaced in[BDR] by the much weaker condition that A have slow dimension growth,i.e., that

fdimΩ n J \lim max < > = 0 .n-+oo j I J )

Since simplicity together with the presence of a unit implies that

lim min {j | Ωn 3\ φ 0} = +oo ,n—» oo '

this condition is automatically fulfilled if the dimensions of the Ωn are uni-formly bounded.

Another condition on the sequence Ax —> A2 —> considered in [BBEK]was small eigenvalue variation. Although this is a condition which is some-what complicated to state compared to the conditions in the preceding para-graph, it is much easier to check in concrete examples of inductive limits,unless one knows for some extraneous reason that the inductive limit has, forexample, a unique trace. It is also a condition which is convenient to workwith in connection with the classification of inductive limits. (See [E110],[Su], [EG], and [BEEK].)

In [BBEK] it was assumed that the spaces Ωn were finite unions of con-nected clopen sets, and hence we have to cast the definition of small eigen-value variation in a slightly different form from that of [BBEK]. In the casethat the Ωn are finite unions of connected clopen sets our present definitionreduces to the previous one.

If x = x* G An and P G 7^(Ωn) then by the eigenvalue list of x (relative toP) we will mean the collection of functions λm on each of the spaces Λ G P ,with λm(ω) the rath lowest eigenvalue of x(ω), counted with multiplicity*Here m = 1,2, , j if Λ C ΩΠfJ .

By the eigenvalue variation of x, denoted by EV(rr), we will mean thequantity

inf max max sup |λm(α;) — λm(α/)| ,

SMALL EIGENVALUE VARIATION AND REAL RANK ZERO 49

where all allowed values of m are considered, namely, m = 1,2, ••• , j ifA C Ω n j . In the case that Ωn is a finite union of connected clopen sets,this infimum is attained at the corresponding partition of Ωn (i.e., the finestpartition), and we recover the definition of [BBEK].

Let us say that A has small eigenvalue variation if

lim EV(Φm n(x)) = 0 for all x = x* G An and all n .m—yoo '

It is clear that if A has real rank zero, then A has small eigenvalue vari-ation. Conversely, it was proved in [BBEK] that if dimΩ n j < 2 for alln and j , and A has small eigenvalue variation, then A has real rank zero.The proof is a simple consequence of the fact (see [CE]) that if Ω is a com-pact metrizable space of dimension at most two, then any function from Ωinto the self-adjoint matrices in Mn can be approximated by a function intothe self-adjoint matrices with nondegenerate spectrum at each point. Thisfails if dimΩ > 3, but, as was first pointed out in [DNNP], the difficultypresented by higher dimensionality can be overcome by use of the Michaelselection theorem. We shall use this theorem in a way which is slightly dif-ferent from the way it was used in [DNNP] (and in [BDR]), namely, toconstruct approximate eigenprojections in C(Ω,Mn).

Using this technique, we have obtained a necessary and sufficient conditionfor real rank zero.

We shall say that A — lim^4n has very small eigenvalue variation if eachalgebra An contains a set Dn of self-adjoint elements such that the union ofthe images of the sets Dn in A is dense in the set of self-adjoint elementsof A (although Dn is not assumed to be dense in the self-adjoint elementsof An), and such that, for any self-adjoint element x of Dn and any ε > 0,there exist m > n and P G V(βm) such that the image h of x in any of thedirect summands C(Λ,M&) of Am with Λ G P has eigenvalue variation atmost ε and, in addition, if [a, b] is any interval of length at least ε inside thespectrum, Sp(/ι), of h then Sp(h(ω)) Π [α,6] contains at least dimΛ points(counted with multiplicity) for each ω G Λ.

A simple argument shows that, in this definition, we may equivalentlyreplace dimΛ by δdimΛ where δ is any fixed positive number. (Break upthe interval of length ε into smaller intervals.)

The reason for introducing this apparently somewhat unwieldy conditionis that not only does it readily follow from the known sufficient conditions^for real rank zero, but in fact it is also necessary. It also turns out to besufficient:

Theorem 1.1. Adopt the general assumptions above on A = limΛn.

The following properties are equivalent.


1. A has very small eigenvalue variation.

2. A has real rank zero.

Before stating a number of corollaries of this result, let us remark that theproof of 2 => 1 is surprisingly simple — in fact, trivial: If A has real rankzero, then Dn may just be taken to be the set of self-adjoint elements in Anwith finite spectrum.

We will defer the proof of 1 => 2 to Section 4.Let us consider various corollaries of Theorem 1.1.

Corollary 1.2. Adopt the general assumptions above on A — \im.An.


1. Each An contains a set Dn of self-adjoint elements such that the union ofthe images of the sets Dn in A is dense in the set of self-adjoint elements ofA, and such that for any n and any self-adjoint element x G Dnj

lim inf max {dim(Λ) EV(ΦΛ Ω (α))} = 0 .


Proof The implication 2 => 1 is trivial just as in Theorem 1.1. To prove1 => 2, we have only to prove that the present condition 1 implies thecondition 1 of Theorem 1.1, with the D n ' s in the definition of very smalleigenvalue variation being the present Z)n's. So, let x = x* £ Dn and ε > 0be given, and use the present condition 1 to choose m such that

d i m ( Ω m > i ) E V ( Φ m i i f n ( a ; ) ) < |

for j = 1, , r m . Fix j , and put Ω = Ωm > j, h = Φmtjtn(x). If dimΩ = 0,then h can be approximated arbitrarily closely by a self-adjoint element withfinite spectrum, and there is nothing to prove. If dim(Ω) > 1, then

i.e., after passing to a possibly finer partition, each eigenvalue in the eigen-value list of h(ω) varies by at most ε/3dimΩ as ω varies over Ω. AsSp(/i) = (J Sp(h(ω)) (without closure since Ω is compact), it follows that, in

order that an interval [α, b] of length at least ε be contained in the spectrumof h (i.e., contain no spectral gap), h(ω) must contain at least

£ - 2 > 3dimΩ - 2 > dimΩ


eigenvalues inside [α, b] for each ω. This shows that A has very small eigen-value variation.

Corollary 1.3. Adopt the general assumptions above on A — lim^4n7 but

assume also that d imΩ m j is bounded uniformly in m and j .


1. A has small eigenvalue variation.


Proof. The implication 1 =Φ- 2 follows immediately from Corollary 1.2 (withDn = A

s

n

Ά), and the implication 2 =» 1 is trivial (cf. [BBEK]).

In the case that A is simple and unital, a stronger form of this result wasproved in [BDR] — with bounded dimension replaced by slow dimensiongrowth (as defined above). (Actually, in place of the condition 1 the resultof [BDR] had the condition that traces should be separated by projections,but in [BBEK] this was shown to be equivalent to the present condition1 — assuming that A is simple and unital, but with no restriction on thedimensions.)

Using Theorem 1.1, we can establish a similar strengthening of Corollary1.3 in the general case (i.e., A not necessarily either simple or unital).

This requires an extension of the definition of slow dimension growth tothe general case. Let us say that A has slow dimension growth (with respectto the sequence A\ -» A2 —> ) if, whenever A is replaced by eAe where eis a projection in Ak for some fc, the condition

still holds.

f d i m Ω n . Ίlim max \ - ^ > = 0π->oo J [ J

Corollary 1.4. Adopt the general assumptions above on A — lim^4n. As-

sume also that A has slow dimension growth {see above).


1. A has small eigenvalue variation.


Proof. 2 => 1 is trivial (cf. [BBEK]).In order to prove 1 =>• 2, we state a lemma which is proved in [Stev]. (In^

the case that A is simple the lemma was proved in [Elll].)

L e m m a 1.5. Adopt the general assumptions above on A — lim^4n. Suppose

that every closed two-sided ideal of A is generated (as an ideal) by its pro-

jections. If n EN, ε > 0 and x — x* G An are given, then there exist Λ > 0


and m0 E N, and P E V(Ωmo), such that for each k E Z and each A E P,if h denotes the image of x in the direct summand of Amo with spectrum A,then either the spectrum of h is disjoint from the interval [kε, (k + l)ε], orat least the fraction δ of the non-zero eigenvalues of h(ω) are contained in[(k - l)ε, (k + 2)ε] for each ω E A.

Let us use this lemma to show that if A has small eigenvalue variation thenA in fact has very small eigenvalue variation, with respect to the sequenceof subsets Dn — A^*".

First, we must show that if A has small eigenvalue variation, then thehypothesis of Lemma 1.5 holds. (This is vacuous in the simple case.) Let/ be a closed two-sided ideal of A, and denote by In the pre-image of I inAn. Let x be a positive element of /n, and let us show that for every ε > 0there exists m > n such that the image of x in Am is within ε of a directsummand of Am contained in Im.

Choose m > n such that the eigenvalue variation of the image of x in Amis strictly less than ε. Then there exists a partition P E V(Ωm) such that,for each A £ P, each eigenvalue of x in the direct summand of Am withspectrum Λ has variation (as a function) at most ε. Hence, either the imageof x in this direct summand is invertible — so that the direct summandbelongs to Im — or it has norm at most ε, as desired.

Let x be a self-adjoint element of An, and let ε > 0 be given. Let δ > 0and ra0 G N be as given by Lemma 1.5. Using small eigenvalue variation,replacing ra0 by a larger number we may suppose that for any m > m0 theimage of x in Am has eigenvalue variation at most ε.

Let Po G 'P(Ωmo) be as given by Lemma 1.5. Using slow dimension growth,choose m>m0 such that, for every Λ E Po, after replacing A by e(A)Ae(A)where e(Λ) E Amo is the unit of the direct summand of Amo with spectrumΛ, we have

dimΩ m jmax : — - < δ .i J

(Of course, we still keep the original A\ we use this form of expression onlyfor convenience.)

Now choose P E 'P(Ωm) such that for every A e Po, the image of the pro-jection e(Λ) E Amo in Am has central support a union of central projectionscorresponding to clopen sets in the partition P.

By the choice of ra0 and Po> for each k E Z and each Λ E Po, with h0 theimage of x in e(Λ)Amo, either the spectrum of h0 is disjoint from the interval[feε, (k + l)ε] or at least the fraction δ of the non-zero eigenvalues of ho(ω)are contained in [(k — l)ε, (k + 2)ε] for each u E Λ .

Fix k E Z. Denote by e E Amo the sum of the projections e(Λ) withA e Po such that the interval [A ε, {k + l)ε] is not disjoint from the spectrum


of the image of x in e(Λ)^4mo. Then the interval [A ε, (k + l)ε\ is disjoint fromthe spectrum of the image of x in (1 — e) Amo (1 — e) and at least the fractionδ of the non-zero eigenvalues of ho{ω) are contained in [(k — l)ε, (k + 2)ε]for each ω in the spectrum of eAmoe, where hQ denotes the image of x ineAmoe. It follows that this is also true with m in place of ra0. (The conditionthat the fraction of non-zero eigenvalues of ho(ω) be at least δ for all ω inthe spectrum of eAmoe (or eAme) is just the condition that all normalizedtraces evaluated on certain continuous functions of h0 be at least δ, andevery normalized trace of eAme restrict to a normalized trace of eAmoe; cf.[Elll], [Stev].)

By the choice of m, the actual number of these non-zero eigenvalues forω in the spectrum of eAme and also in Ωm>J is at least

δj > dimΩm 5 J .

This shows that, for every Λx E P, for every k E Z, either the interval[A ε, (A; + l)ε] is disjoint from the spectrum of the image h of x in the directsummand of Am with spectrum Λi, or else at least dimΛi eigenvalues ofh(ω) are contained in [(A; — l)ε, (A; -f 2)ε] for every ω E Ax. (With j such thatΩ m j 2 Λi, one has d i m Ω m j > dimΛi.)

Hence, for any Λi E P, if [α, 6] is any interval of length 4ε inside thespectrum of h (necessarily containing some interval [(A; — l)ε, (A; + 2)ε]),Sp(/ι(α;)) Π [α,6] contains at least dimΛi points (counting multiplicity).

This shows that A has very small eigenvalue variation. Hence by Theorem1.1, A has real rank zero.

2. Connectedness of frame spaces.

Let Vjfc(Cn) denote the Stiefel manifold of A -orthogonal frames in C n and

Gfc(Cn) the Grassmannian of A -dimensional subspaces of C71. Consider the

canonical map

which to each frame associates the linear span of the vectors in it.

Let Cp C O1 denote the p-dimensional supspace of C^ spanned by thep first coordinate vectors, and let C n " p C C71 denote the subspace spannedby the n — p last coordinate vectors. Assume that p < k < n. There is aninclusion

given by ip(V) =€P ®V. Set

lmip}


Thus, B is the space of /j-frames whose associated subspace contains C \Let 7Γi denote the zth homotopy group.

Lemma 2.1. π^B) — 1 for

i < 2 min(k — p,n — k) .

Proof. The standard frame in Cp, combined with a (k — p)-frame in Cn~p,gives a λ -frame in 5 . Thus, the natural fibration

- p ) -+ V,

maps in a natural way into the fibration

where the map B —> Gi_p(Cn~p) consists in taking the orthogonal comple-ment of Cp in τr(F) for F € B.

The map Gfc_p(C"-p) ->• Gfc_p(C

n-p) is the identity.In the associated map of long exact sequences

every third vertical map is therefore an isomorphism.If i < 2{k - p), by Theorem 1.7.4.1 of [Hus] the map ^(U(A; - p)) ->

τrj(U(fc)) is also an isomorphism, and hence by the five lemma the mapπι(Vk^p(C

n'p)) -> Έi(B) is an isomorphism. By Theorem 1.7.5.1 of [Hus],the group πi(Vk^p(C

ri-p)) is trivial if i < 2((n-p) - (k-p)) = 2(n - fc). Itfollows that πτ(B) is trivial for i < 2min(A: — p,n — k).

3. The selection theorem.

If Ω is a compact Hausdorff space, and F is a map from Ω into the space ofself-adjoint elements of Mn, we shall say that F is upper semicontinuous iffor every vector ( E C the real-valued function

ω H+ (F(ω)ξ I ξ)

is upper semicontinuous. We shall say that F is lower semicontinuous ifthese functions are lower semicontinuous.


Clearly, F is continuous if and only if F is both upper and lower semicon-tinuous.

If x = x* E C(Ω,Mn), i.e., x is a continuous function from Ω into theself-adjoint elements of Mn, and P(] — oo,λ[) and P(] — oc,λ]) denote thespectral projections of x corresponding to λ E K, then ω •-> Pi] — oo, λ[)(ω)is lower semicontinuous, as the corresponding characteristic function is lowersemicontinuous, and, similarly, ω H» P(] — OO, λ])(α;) is upper semicontinuous.

Our selection theorem is as follows.

Theorem 3.1. Let Ω be a compact metrizable Hausdorff space of dimensiond, and let P and Q be maps from Ω into the projections of Mn such that Pis lower semicontinuous and Q is upper semicontinuous. Suppose that

P(ω) > Q(ω)

for each ω E Ω, and that, furthermore, there exists a natural number k suchthat

dimP(α ) > k +-(d + 1)

for all ω E Ω, and

dimQ(α ) i?(ω), from Ω into

the k-dimensional projections of Mn, such that

P(ω) > R{ω) > Q(ω)

for all ω E Ω.

Proof We will partly follow [DNNP] and use Michael's selection theorem,Theorem 1.2 of [Mi], which states: If Ω is a compact metrizable space ofdimension d, T is a complete metric space, and Y is a map from Ω to thenonempty closed subsets of T such that

(a) Y is lower semicontinuous, i.e., for each open subset U of T the set{ω E Ω I Y(ω) Π U Φ 0} is open,

(b) each Y(ω) is (d + l)-connected, i.e., ni(Y(ω)) = 0 for i = 0,1, • ,d + 1, and

(c) there is an ε > 0 such that for any 0 < r < ε and ω E Ω, the-intersection of Y(ω) with any closed ball of radius r in T is a contractiblespace,

then there is a continuous map p : Ω —> T such that p(ω) E Y{ω) for allω E Ω.


We will apply this selection theorem in the case T = Vk (Cn) = the Stiefel

manifold of A -orthogonal frames in Cn, where k, n are as in the statement ofthe theorem, and

Y(ω) = {F G T I Q(ω) < π{F) < P{ω)} .

Let us verify the hypotheses of Michael's selection theorem.Clearly, each Y(ω) is a nonempty closed subset of T.

Ad (a). If U is an open subset of V*;(Cn), then π(U) is an open subsetof the set of λ -dimensional projections on C1. Since Ω *-+ Q(ω) is uppersemicontinuous and ω ι-» P(ω) is lower semicontinuous, it follows that theset

{ω G Ω I 3q G π(U), Q(ω) \{d + 1), it follows in particular that πi(Y(ω)) = 0 for 0 < i < d + 1,and so (b) holds.

Ad (c). The local connectivity of Y{ω) is trivial in our case.Applying Michael's selection theorem, we obtain a continuous map F :

Ω-> V*(Cn). SetR(ω) =π(F(ω)) .

Then ω H-> R(ω) is a continuous map from Ω into the self-adjoint A -dimensionalprojections in Mn such that

P(ω) > R{ω) > Q{ω)

for all ω G Ω, as desired.

4. Sufficiency of very small eigenvalue variation.

In this section we shall prove the remaining implication 1 => 2 in Theorem1.1. Suppose that A has very small eigenvalue variation, and let x — x* E A.We wish to approximate a: by a self-adjoint element with finite spectrum,and to begin with we may approximate x by an element of some Dn (where


Dn is as specified just before the statement of Theorem 1.1). Thus, we mayassume that x G Dn. By the definition of very small eigenvalue variation,for any ε > 0 there is an m > n such that the image h of x in any of thedirect summands C(Ω,MΛ) of Am has eigenvalue variation strictly less thanε/3, and, in addition, if [α, b] is any interval of length at least ε/3 inside thespectrum of /ι, then Sp(/ι(α;)) Π [α, 6] contains at least dim(Ω) + 2 points.(This is provided dimΩ φ 0; in the trivial case dimΩ = 0, h may easily beapproximated by elements of finite spectrum.) We need to approximate hby an element y with finite spectrum. Choose a G K and j Έ N such thatSp(/ι) C [α, a + jε], and for each i — 1,2, , j set

p̂ ~ = spectral projection of /ι

corresponding to the interval

] -oo,α + ( i - l)ε] ,

Pf — spectral projection of h

corresponding to the interval

] — oo,α + iε[.

Thus,

PΓ < Pi < ?2 < < K ,

the map ω ι-> i:>ί~(u;) ^s upper semicontinuous from Ω into the space of

projections in Mfc, and the map ω ι-> Pt(ω) is lower semicontinuous.

Given i G {1,2, ,7}, there are two possibilities.

Case 1. [α + (i - l)ε + ε/3, α + iε - ε/3] £ Sp(/ι).In this case, choose λ in this interval which is not in Sp(/ι), and consider

the spectral projection Pi of h corresponding to the interval ] — 00, λ]. Thenω H-» Pi(ω) is continuous, i.e., P{ G C(Ω,Mfc), and

PΓ < Pi < Pt

Case 2. / := [α + (i - l)ε + ε/3 , a + iε - ε/3] C Sp(Λ).In this case, Sp(h(ω)) Π / contains at least dim(Ω) 4- 2 points for each

ω G Ω. But since each eigenvalue of /ι(α ) varies by strictly less than ε/3 asω varies over Ω, it follows that there exists an integer I such that

dimi^-(cc ) / + ^(dim(Ω) + 1)


for each ω G ί l . Prom Theorem 3.1 (the selection theorem), it follows thatthere exists a continuous map ω E Ω »-> Pι(ω) from Ω into the /-dimensionalprojections in Mk such that

PΓ(ω)


References

[BBEK] B. Blackadax, O. Bratteli, G. A. Elliott, and A. Kumjian, Reduction of real rank ininductive limits of C*-algebras, Math. Ann., 292 (1992), 111-126.

[BDR] B. Blackadax, M. Dadarlat, and M. R0rdam, The real rank of inductive limit C-algebras, Math. Scand., 69 (1991), 211-216.

[BEEK] O. Bratteli, G.A. Elliott, D.E. Evans, and A. Kishimoto, On the classification ofC* -algebras of real rank zero, III: The infinite case, Operator Algebras and theirApplications (editors, P.A. Fillmore and J.A. Mingo), Fields Institute Communica-tions, American Mathematical Society, to appear.

[CE] M.-D. Choi and G.A. Elliott, Density of the self-adjoint elements with finite spec-trum in an irrational rotation C*-algebra, Math. Scand., 167 (1990), 73-86.

[DNNP] M. Dadarlat, G. Nagy, A. Nemethi, and C. Pasnicu, Reduction of topological stablerank in inductive limits of C*-algebras, Pacific J. Math., 153 (1992), 267-276.

[E110] G.A. Elliott, On the classification of C*-algebras of real rank zero, J. Reine Angew.Math., 443 (1993), 179-219.

[Elll] , A classification of certain simple C*-algebras, Quantum and Non-Commu-tative Analysis (editors, H. Araki et al.), Kluwer, Dordrecht, 1993, 373-385.

[EG] G.A. Elliott and G. Gong, On the classification of C*-algebras of real rank zero, II,Ann. of Math., to appear.

[Eng] R. Engelking, Dimension Theory, North Holland, Amsterdam, 1978.[Hus] D. Husemoller, Fibre Bundles, McGraw-Hill, New York, 1966.[Mi] E. Michael, Continuous selections II, Ann. of Math., 164 (1956), 562-580.

[Stev] K.H. Stevens, The classification of certain non-simple approximate interval algebras,Ph.D. thesis, University of Toronto, 1994.

[Su] H. Su, On the classification of C* -algebras of real rank zero: inductive limits ofmatrix algebras over non-Hausdorff graphs, Ph.D. thesis, University of Toronto,1992, and Memoirs Amer. Math. Soc, 547 (1995).

Received January 5, 1994.

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Volume 175 No. 1 September 1996

1Homogeneous Ricci positive 5-manifoldsDIMITRI ALEKSEEVSKY, ISABEL DOTTI DE MIATELLO and CARLOS J.FERRARIS

13On the structure of tensor products of `p-spacesALVARO ARIAS and JEFFREY D. FARMER

39The closed geodesic problem for compact Riemannian 2-orbifoldsJOSEPH E. BORZELLINO and BENJAMIN G. LORICA

47Small eigenvalue variation and real rank zeroOLA BRATTELI and GEORGE A. ELLIOTT

61Global analytic hypoellipticity of �b on circular domainsSO-CHIN CHEN

71Sharing values and a problem due to C. C. YangXIN-HOU HUA

83Commutators and invariant domains for Schrödinger propagatorsMIN-JEI HUANG

93Chaos of continuum-wise expansive homeomorphisms and dynamical properties ofsensitive maps of graphs

HISAO KATO117Some properties of Fano manifolds that are zeros of sections in homogeneous vector

bundles over GrassmanniansOLIVER KÜCHLE

127On polynomials orthogonal with respect to Sobolev inner product on the unit circleXIN LI and FRANCISCO MARCELLAN

147Maximal subfields of Q(i)-division ringsSTEVEN LIEDAHL

161Virtual diagonals and n-amenability for Banach algebrasALAN L. T. PATERSON

187Rational Pontryagin classes, local representations, and K G -theoryCLAUDE SCHOCHET

235An equivalence relation for codimension one foliations of 3-manifoldsSANDRA SHIELDS

257A construction of Lomonosov functions and applications to the invariant subspaceproblem

ALEKSANDER SIMONIČ271Complete intersection subvarieties of general hypersurfaces

ENDRE SZABÓ

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athematics

1996Vol.175,N

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Paciﬁc Journal of Mathematics - MSP · PACIFIC JOURNAL OF MATHEMATICS Vol. 175, No. 1, 1996 SMALL EIGENVALUE VARIATION AND REAL RANK ZERO OLA BRATTELI AND GEORGE A. ELLIOTT A necessary